Question

The dataframe nlmodel contains data on a response variable y and a single explanatory variable x. A scatter plot of y versus x suggests a strong non-linear relationship: 200 ismicdie 150 100 - 50 0.00 0.25 0.75 1.00 0.50 X Suppose for these data we wish to consider the model 012 YiN 02 + Ii i=1 (a) [1 mark] Why can’t this model be fit using a linear (regression) model? (b) [2 marks] Write down the likelihood L(θ1, θ2, σ2; y, x) and the log-likelihood `(θ1, θ2, σ2; y, x) 1(c) [1 mark] Write an R function mylike() which evaluates the negative log-likelihood (i.e. −`(θ1, θ2, σ; y, x)) for any values of the three parameters (d) [5 marks] Use the R function nlm() in association with your function mylike() to numerically minimise the log-likelihood. Provide some evidence of how you chose sensible starting values. Report the maximum likelihood estimates of the parameters and superimpose a plot of the associated mean relationship on a scatter plot of y versus x. (e) [5 marks] Report the standard errors for θ1 and θ2, and use those to construct 95% confidence intervals. (f) [3 marks] Test the hypothesis that θ2 = 0.08 at the 5% significance level (not using the confidence interval) and compute the associated p-value of the test. (g) [4 marks] Use plug-in prediction to construct and plot 95% prediction intervals. Show transcribed data 200 ismicdie 150 100 - 50 0.00 0.25 0.75 1.00 0.50 X Suppose for these data we wish to consider the model 012 YiN 02 + Ii i=1,2,..., 100, Y; independent 1:0) 02

Answer 1

A scatter plot with a non-linear pattern indicates that a linear regression model is not suitable. One should consider non-linear models or data transformations. In statistical analysis, R's nlm() function can be used to find the best model parameters through the minimization of the negative log-likelihood.

Step-by-step explanation:

When examining a scatter plot, it is important to identify the pattern of the data points. If the data suggest a linear relationship, then a linear regression model might be suitable. However, a non-linear pattern suggests that linear regression isn't appropriate. In such cases, it is necessary to consider non-linear models or transformations that can linearize the relationship for analysis.

For the likelihood and log-likelihood, these are fundamental concepts in statistics used to estimate the parameters of a statistical model. They measure how well the model explains the observed data. The negative log-likelihood is often used in optimization problems because it turns the maximization problem into a minimization one, which is more common in optimization algorithms.

Using R for statistical computing, the nlm() function can be utilized to numerically minimize the negative log-likelihood, thus finding the parameter estimates that make the observed data most probable. Moreover, once the model parameters are estimated, standard errors can be calculated to construct confidence intervals and perform hypothesis testing to evaluate the significance of the parameters.